Tensor product \(q\)-Bernstein polynomials (Q1014883)

Given a real number \(q>0\), \(q\)-Bernstein polynomials are a generalization, in the spirit of \(q\)-calculus, of the classical Bernstein polynomials (which can be obtained for \(q=1\)) where some of the integers in the definition of the classical ones are substituted by \(q\)-integers. For \(q \neq 1\) the properties of \(q\)-Bernstein polynomials differ essentially from those in the classical case. (See the recent paper by \textit{G. M. Phillips} [A survey of results on the \(q\)-Bernstein polynomials, accepted in IMA J. Numer. Analysis (2009)] for a detailed account of the state-of-art in the subject.) In a preview paper by \textit{H. Oruç} and \textit{G. M. Phillips} [J. Comput. Appl. Math. 151, No.~1, 1--12 (2003; Zbl 1014.65015)], the notion of \(q\)-Bernstein-Bézier curves is studied. In the paper under review the authors present, as a continuation of preview works, a new de Casteljau type algorithm to compute a \(q\)-Bernstein curve different from a previous one also obtained by Phillips but that enables the authors to construct tensor product \(q\)-Bernstein surfaces and to analyze their fundamental properties.